 Solution

The topology on $A$ is the subspace topology inherited from the complex plane. A subset $B$ is open in $A$ if $B=A\cap U$ for some open subset $U\subseteq \mathbb{C}$. Equivalently, every $b\in B$ has a radius $ϵ>0$ such that $A\cap \{z:|z-b|<ϵ\}\subseteq B$.

A subset $B$ is closed in $A$ if $A\setminus B$ is open in $A$, equivalently if $B=A\cap F$ for a closed subset $F$ of the complex plane. These are relative properties: the same subset need not be open or closed in the ambient plane.